Subdiffusion in a system consisting of two different media separated by a thin membrane

Abstract

We consider subdiffusion in a system which consists of two media separated by a thin membrane. The subdiffusion parameters may be different in each of the medium. Using the new method presented in this paper we derive the probabilities (the Green's functions) describing a particle's random walk in the system. Within this method we firstly consider the particle's random walk in a system with both discrete time and space variables in which a particle can vanish due to reactions with constant probabilities R1 and R2 defined separately for each medium. Then, we move from discrete to continuous variables. The reactions included in the model play a supporting role. We link the reaction probabilities with the other subdiffusion parameters which characterize the media by means of the formulae presented in this paper. Calculating the generating functions for the difference equations describing the random walk in the composite membrane system with reactions, which depend explicitly on R1 and R2, we are able to correctly incorporate the subdiffusion parameters of both the media into the Green's functions. Finally, placing R1 = R2 = 0 into the obtained functions we get the Green's functions for the composite membrane system without any reactions. From the obtained Green's functions, we derive the boundary conditions at the thin membrane. One of the boundary conditions contains the Riemann-Liouville fractional time derivatives, which show that the additional 'memory effect' is created by a discontinuoity of the system. The second boundary condition demands continuity of a flux at the membrane. keywords: subdiffusion, diffusion in composite systems with a thin membrane, random walk model

Modelling experimentally measured of ciprofloxacin antibiotic diffusion in Pseudomonas aeruginosa biofilm formed in artificial sputum medium

“…Unfortunately, the statistic is too poor to determine the error for α . Because the consistency of ASM is very similar to the consistency of 1% aqueous agarose solution for which α = 0.95 [ 40 ], we suppose that there is subdiffusion in the ASM medium and the value of the parameter α is realistic. We note that anomalous diffusion exponents as about of 0.9 are not exceptional and physically meaningful, see e.g.…”

Model of ciprofloxacin subdiffusion inPseudomonas aeruginosabiofilm formed in artificial sputum medium

“…Unfortunately, the statistic is too poor to determine the error for α. Because the consistency of ASM is very similar to the consistency of 1% aqueous agarose solution for which α = 0.95 [9], we suppose that there is subdiusion in the ASM medium and the value of the parameter α is realistic.…”

Boundary conditions at a thin membrane for normal diffusion, classical subdiffusion, and slow subdiffusion processes

“…In our opinion, this question should remain open, although the following facts prove the model usefulness: (1) The model has a simple physical interpretation, (2) the model provides equations of different types of diffusion that can also be derived from the other models such as the CTRW model, 4,6 (3) the boundary condition derived from this model for normal diffusion (49) has been confirmed empirically, 38 and (4) the boundary condition for classical subdiffusion (53) have provided solutions to the subdiffusion equation that coincide with the empirical results describing the release of antibiotics from the gel medium through a partially permeable membrane. 30 However, regardless of the above arguments, the following conclusion seems to be universal: the boundary conditions at the membrane should take into account the selective properties of the membrane as well as the type of diffusion in the system.…”

“…29 The model presented in this paper has been used to model different diffusion processes in systems with a thin membrane. 18,[30][31][32][33][34] The aim of this paper is to show that boundary conditions at the membrane should depend on the type of diffusion. We also present the new form of boundary condition at the membrane for the slow subdiffusion process.…”

“…We mention that such a way of deriving boundary conditions at fully reflecting or fully absorbing wall for normal diffusion has been presented in Chandrasekhar 29 . The model presented in this paper has been used to model different diffusion processes in systems with a thin membrane 18,30-34 . The aim of this paper is to show that boundary conditions at the membrane should depend on the type of diffusion.…”